Let Y1 < Y2 < : : : < Y9 be the order statistics of 9 independent draws from an
exponential distribution that has a mean of 2.
(1) Find the PDF of Y2.
(2) Compute P[Y9 < 1].
From 6.3-3. Let Y1 < Y2 < . . . < Y9 be the order statistics of 9 independent draws from an exponential distribution that has a mean of 2. (1) Find the PDF of Y2. (2) Compute P[Y9 < 1]. From 6.3-3. Let Yǐ < ½ < . . . < y) be the order statistis of 9 independent draws from an exponential distribution that has a mean of 2. (1) Find the PDF of Y2 2) Compute PY,1
From 6.3-3. Let Yi < Y2 < < Yg be the order statistics of 9 independent draws from an exponential distribution that has a mean of 2. (1) Find the PDF of Y2 (2) Compute PIY9 < 1
Let Y1< Y2< Y3< Y4< Y5 be the order statistics of n=5 independent observations from the exponential distribution with mean= 1. determine P(Y1>1) and find the pdf of Y5
. Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from an exponential distribution with parameter θ = 1. (a) Find the pdf of Yr. (b) Find the pdf of U = e −Yr .
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following questions. (a) decide whether Z1 = Y2 and Z2=Y4-Y2 are stochastically independent or not. (hint. first find the joint p.d.f. of Y2 and Y4) (b) show that Z1 = nY1, Z2= (n-1)(Y2-Y1), Z3=(n-2)(Y3-Y2), ...., Zn=Yn-Yn-1 are stocahstically independent and that each Zi has the exponential distribution.(hint use change of variable technique)
Let Y1, Y2, . . . , Yn be independent random variables with Exponential distribution with mean β. Let Y(n) = max(Y1,Y2,...,Yn) and Y(1) = min(Y1,Y2,...,Yn). Find the probability P(Y(1) > y1,Y(n) < yn).
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3] 15. (30 points) Let Y1
Let Y1, Y2 have the joint density f(y1,y2) = 4y1y2 for 0 ≤ y1,y2 ≤ 1 = 0 otherwise (a) (8 pts) Calculate Cov(Y1, Y2). (b) (3 pts) Are Y1 and Y2 are independent? Prove your answer rigorously. (c) (6 pts) Find the conditional mean E(Y2|Y1 = 1). 3
Let Y1, Y2,. . , Yn be a random sample from the population with pdf f(u:)elsewhere (a) If WIn Yi, show that W, follows an exponential distribution with mean 1/0. (b) Show that 2θΣηι W, follows a χ2 distribution with 2n degrees of freedom. (c) It turns out that if X2 distribution with v degrees of freedom, then E( Use this to show
Let Y1, Y2, …, Y4 be a random sample from a normal distribution with mean 10 years and standard deviation 2.5 years. Find the following probabilities. A. P(Y4 > 14 years) B. P(Y1 + Y2 + Y3 + Y4 < 36 years) C. P{(Y1 < 9 years) and (Y2 < 9 years) and (Y3 < 9 years) and (Y4 < 9 years)} Note: B and C are asking different questions. D. Find E(Y1 +...